Abstract

We review recent progress towards a rigorous understanding of the Bogoliubov approximation for bosonic quantum many-body systems. We focus, in particular, on the excitation spectrum of a Bose gas in the mean-field (Hartree) limit. A list of open problems will be discussed at the end.

Received 15 October 2013Accepted 02 April 2014Published online 26 June 2014

Acknowledgments:

Partial financial support by the NSERC is gratefully acknowledged.

Article outline:I. INTRODUCTIONII. THE BOSE GAS: A QUANTUM MANY-BODY PROBLEMIII. THE IDEAL BOSE GASIV. SECOND QUANTIZATION ON FOCK SPACEV. THE BOGOLIUBOV APPROXIMATIONVI. VALIDITY OF THE BOGOLIUBOV APPROXIMATIONVII. THE MEAN-FIELD (HARTREE) LIMITA. Main resultsB. Ideas in the proof1. Step 1: Approximation by a quadratic Hamiltonian2. An algebraic identity3. Step 2: The spectrum of C. Conclusions and generalizationsVIII. OPEN PROBLEMS